Uniqueness of signature for simple curves

2013/04/02 by H. Boedihardjo, Boedihardjo, H., H. Ni +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1304.0755

Results extended to cover the case of paths with finite p variation, p between 1 and 2, and formulated in terms of log signature

arxiv created 2014/07/15 · arxiv updated 2014/07/17

Abstract

We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1≤ p<2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLE_κ null set, where 0<κ≤ 4, the signature series of curves uniquely determine the curves. Our calculations also enable us to express the Fourier transform of the n-point functions of SLE curves in terms of the expected signature of SLE curves. Although the techniques used in this article are deterministic, the results provide a platform for studying SLE curves through the signatures of their sample paths.

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